A056608 -id:A056608 - cp's OEIS frontend

A141284 a(n) = (p_max - 1)...p...(p_min + 2), where p_max...p...p_min = k(n) = n-th composite.

4, 8, 8, 10, 16, 16, 24, 20, 16, 24, 32, 30, 40, 32, 28, 48, 30, 48, 48, 32, 50, 64, 42, 48, 72, 60, 64, 72, 80, 60, 88, 64, 54, 80, 80, 96, 72, 70, 96, 90, 112, 96, 120, 90, 64, 84, 120, 128, 110, 120, 96, 144, 100, 144, 90, 144, 128, 90, 160, 144, 112, 168, 140, 160

Offset: 1

Juri-Stepan Gerasimov, Aug 08 2008

nonn

In the prime factorization of the n-th composite, replace one instance of the largest prime factor A052369(n) with A052369(n)-1 and replace one instance of the smallest prime factor A056608(n) with A056608(n)+2.

			For n=1, k(1) = 4 = (p_max=2)*(p_min=2), so a(1) = (2-1)*(2+2) = 1*4 = 4;
for n=2, k(2) = 6 = (p_max=3)*(p_min=2), so a(2) = (3-1)*(2+2) = 2*4 = 8;
for n=3, k(3) = 8 = (p_max=2)*(p=2)*(p_min=2), so a(3) = (2-1)*2*(2+2) = 1*2*4 = 8; etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A002808.

Map[Times @@ Flatten[{#[[1]] + 2, #[[2 ;; -2]], #[[-1]] - 1}] &@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#]] &, Select[Range[120], CompositeQ]] (* Michael De Vlieger, Jan 25 2023 *)

a(n) = A002808(n)*(A052369(n)-1)*(A056608(n)+2)/(A052369(n)*A056608(n)).

Definition and examples corrected and entries checked by R. J. Mathar, Mar 29 2010

Simplified by Jon E. Schoenfield, Jan 25 2023

A161986 a(n) = k+r where k is composite(n) and r is (largest prime divisor of k) mod (smallest prime divisor of k).

Original entry on oeis.org

4, 7, 8, 9, 11, 13, 15, 17, 16, 19, 21, 22, 23, 25, 25, 27, 27, 29, 31, 32, 35, 35, 37, 37, 39, 40, 41, 43, 45, 47, 47, 49, 49, 51, 53, 53, 55, 56, 57, 58, 59, 61, 63, 64, 64, 68, 67, 69, 71, 71, 73, 75, 77, 77, 81, 79, 81, 81, 83, 85, 87, 87, 89, 89, 91, 97, 93, 94, 95, 99, 97

Offset: 1

Klaus Brockhaus, Jun 23 2009

nonn

Auxiliary sequence for A161850, which is the subsequence consisting of all terms that are prime.

a(n) = A002808(n)+A161849(n).

			n = 1: composite(1) = 4; (largest prime divisor of 4) = (smallest prime divisor 4) = 2; 2 mod 2 = 0. Hence a(1) = 4+0 = 4.
n = 5: composite(5) = 10; (largest prime divisor of 10) = 5; (smallest prime divisor 10) = 2; 5 mod 2 = 1. Hence a(5) = 10+1 = 11.

Bill McEachen, Table of n, a(n) for n = 1..10000

Cf. A161850, A002808 (composite numbers), A052369 (largest prime factor of n-th composite), A056608 (smallest divisor of n-th composite), A161849 (A052369(n) mod A056608(n)).

[ n + D[ #D] mod D[1]: n in [2..100] | not IsPrime(n) where D is PrimeDivisors(n) ];

genit(maxx=1000)={ctr=0;arr=List();forcomposite(k=4,+oo,v=factor(k)[,1];r=v[#v]%v[1];ctr+=1;if(ctr>=maxx,break);listput(arr,k+r));arr} \\ Bill McEachen, Nov 17 2021

A082049 Least composite number greater than n-th composite number having greater smallest prime factor than that of n-th composite number.

Original entry on oeis.org

9, 9, 9, 25, 15, 15, 15, 25, 21, 21, 21, 25, 25, 25, 49, 27, 35, 33, 33, 33, 35, 35, 49, 39, 39, 49, 45, 45, 45, 49, 49, 49, 121, 51, 55, 55, 55, 77, 57, 65, 63, 63, 63, 65, 65, 77, 69, 69, 77, 75, 75, 75, 77, 77, 121, 81, 81, 85, 85, 85, 91, 87, 91, 91, 91, 121, 93, 95, 95

Offset: 1

Reinhard Zumkeller, Apr 02 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002808, A020639, A056608.

seq[lim_] := Module[{c = Select[Range[lim], CompositeQ], p, s = {}, n, v}, p = FactorInteger[#][[1, 1]] & /@ c; n = Length[p]; Do[v = 0; Do[If[p[[j]] > p[[i]], v = c[[j]]; Break[]], {j, i + 1, n}]; If[v == 0, Break[], AppendTo[s, v]], {i, 1, n}]; s]; seq[125] (* Amiram Eldar, Mar 26 2025 *)

A020639(a(n)) > A020639(A002808(n)).

A082048(A002808(n)) <= a(n).

A118663 Index of the least prime dividing the n-th composite number: A000720(A020639(A002808(n))).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1, 2, 1, 1, 4, 1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1, 2, 1, 4, 1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 4, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 2, 1, 4, 1, 5, 1, 2, 1, 3, 1, 1, 2, 1, 1, 4, 1, 2, 1, 1, 1, 2

Offset: 1

Giovanni Teofilatto, May 19 2006

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A056608.

PrimePi[FactorInteger[#][[1,1]]]&/@Select[Range[200],CompositeQ] (* Harvey P. Dale, May 13 2023 *)

for(i=2,999,isprime(i)||print1(primepi(factor(i)[1,1])",")) \\ - M. F. Hasler, Apr 03 2012

A118663 = A000720 o A056608. - M. F. Hasler, Apr 03 2012

A141554 Transformed nonprime products of prime factors of the composites, the largest prime decremented by 2 and the smallest prime incremented by 2.

Original entry on oeis.org

0, 4, 0, 12, 8, 20, 15, 0, 12, 24, 25, 36, 16, 21, 44, 15, 40, 36, 0, 45, 60, 35, 24, 68, 55, 48, 60, 72, 45, 84, 32, 45, 60, 75, 88, 36, 63, 80, 85, 108, 72, 116, 75, 0, 77, 108, 120, 105, 100, 48, 140, 75, 136, 81, 132, 96, 45, 156, 120, 105, 164, 135, 144, 108, 99, 168

Offset: 1

Juri-Stepan Gerasimov, Aug 14 2008

nonn

In the prime number decomposition of k=A002808(i), i=1,2,3,.., one instance of the largest prime, pmax=A052369(i), is replaced by pmax-2 and one instance of the smallest prime, pmin=A056608(i), is replaced by pmin+2. If the product of this modified list of factors, k*(pmax-2)*(pmin+2)/(pmin*pmax), is nonprime, it is added to the sequence.

			k(1)=4=(p(max)=2)*(p(min)=2), transformed (2-2)*(2+2)=0*4=0=a(1).
k(2)=6=(p(max)=3)*(p(min)=2), transformed (3-2)*(2+2)=1*4=4=a(2).
k(3)=8=(p(max)=2)*(p=2)*(p(min)=2), transformed (2-2)*2*(2+2)=0*2*4=0=a(3).
k(4)=9=(p(max)=3)*(p(min)=3), transformed (3-2)*(3+2)=1*5=5 (prime, skipped).
k(5)=10=(p(max)=5)*(p(min)=2), transformed (5-2)*(2+2)=3*4=12=a(4).

Cf. A141218, A141219, A141220, A141284.

Edited and corrected by R. J. Mathar, Aug 18 2008

A161850 Subsequence of A161986 consisting of all terms that are prime.

Original entry on oeis.org

7, 11, 13, 17, 19, 23, 29, 31, 37, 37, 41, 43, 47, 47, 53, 53, 59, 61, 67, 71, 71, 73, 79, 83, 89, 89, 97, 97, 101, 101, 103, 107, 109, 113, 127, 131, 137, 137, 139, 149, 149, 151, 157, 163, 163, 167, 167, 173, 179, 179, 181, 193, 191, 193, 197, 199, 211, 223, 227

Offset: 1

Juri-Stepan Gerasimov, Jun 20 2009

nonn

A161986(n) = k+r where k is n-th composite and r is remainder of (largest prime divisor of k) divided by (smallest prime divisor k).

			A161986(1) to A161986(27) are 4, 7, 8, 9, 11, 13, 15, 17, 16, 19, 21, 22, 23, 25, 25, 27, 27, 29, 31, 32, 35, 35, 37, 37, 39, 40, 41. Hence a(1) to a(11) are the prime terms among them, namely 7, 11, 13, 17, 19, 23, 29, 31 ,37, 37, 41.

Cf. A161986 (A002808(n)+A161849(n)), A002808 (composite numbers), A161849 (A052369(n) mod A056608(n)), A052369 (largest prime factor of n-th composite), A056608 (smallest divisor of n-th composite).

[ p: n in [2..230] | not IsPrime(n) and IsPrime(p) where p is n+D[ #D] mod D[1] where D is PrimeDivisors(n) ];

Edited and corrected (a(19)=57 replaced by 67; a(38)=137, a(49)=179, a(50)=179 inserted) by Klaus Brockhaus, Jun 24 2009

A242016 Smallest prime factor of A135972(n), the n-th composite Mersenne number.

Original entry on oeis.org

3, 3, 3, 7, 3, 23, 3, 3, 7, 3, 3, 3, 7, 3, 47, 3, 31, 3, 7, 3, 233, 3, 3, 7, 3, 31, 3, 223, 3, 7, 3, 13367, 3, 431, 3, 7, 3, 2351, 3, 127, 3, 7, 3, 6361, 3, 23, 3, 7, 3, 179951, 3, 3, 7, 3, 31, 3, 193707721, 3, 7, 3

Offset: 1

Felix Fröhlich, Aug 11 2014

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..1201

Cf. A135972, A242017. Subsequence of A056608.

for(n=2, 1e2, if(!ispseudoprime(2^n-1), p=factor(2^n-1)[1, 1]; print1(p, ", ")))

a(n) = A020639(A135972(n)).

A085434 Twice odd isolated primes.

Original entry on oeis.org

46, 74, 94, 106, 134, 158, 166, 178, 194, 226, 254, 262, 314, 326, 334, 346, 422, 446, 466, 502, 514, 526, 554, 586, 614, 634, 662, 674, 706, 718, 734, 746, 758, 766, 778, 794, 802, 818, 878, 886, 898, 914, 934, 958, 974, 982, 998, 1006, 1018, 1082, 1094

Offset: 6

Cino Hilliard, Aug 13 2003

Name was: n-th even number not a power of 2 whose largest and smallest factors do not add or subtract to a twin prime. - Robert Israel, Mar 11 2025

The density of these numbers approach 0 as n approaches oo.

Cf. A052369, A056608, A134797.

P:= select(isprime, {seq(i,i=3..1000,2)}):
A:= P minus (P +~ 2) minus (P -~ 2):
sort(convert(A,list)) *~ 2; # Robert Israel, Mar 11 2025

maxpmmintwin(n) = { c=0; forprime(p=3,n, if(!isprime(p-2) & !isprime(p+2),print1(p+p","); c++); ); print(); print(c" "c/n+.0) }

a(n) = 2 * A134797(n).

Definition corrected by Robert Israel, Mar 11 2025

A091094 Number of partitions of n-th composite number not containing the smallest prime factor.

Original entry on oeis.org

3, 6, 11, 19, 20, 35, 58, 99, 96, 154, 242, 407, 375, 573, 1331, 861, 1435, 1282, 1886, 2745, 4539, 3961, 9279, 5667, 8038, 13208, 11323, 15836, 22001, 35960, 30383, 41715, 120351, 56953, 92670, 77363, 104566, 247050, 140668, 227999, 188397

Offset: 1

Reinhard Zumkeller, Feb 22 2004

nonn

a(n) = A000041(A002808(n)) - A091114(n).

a(n) = A000041(A002808(n)) - A000041(A085271(n)). - Charlie Neder, Jan 11 2019

Cf. A002808, A000041, A020639, A056608.

Incorrect formula removed by Charlie Neder, Jan 11 2019

A141553 Transformed nonprime products of prime factors of the composites, the largest prime decremented by 2 and the smallest prime incremented by 1.

Original entry on oeis.org

0, 0, 4, 9, 6, 15, 12, 0, 9, 18, 20, 27, 12, 18, 33, 12, 30, 27, 0, 36, 45, 30, 18, 51, 44, 36, 45, 54, 36, 63, 24, 40, 45, 60, 66, 27, 54, 60, 68, 81, 54, 87, 60, 0, 66, 81, 90, 84, 75, 36, 105, 60, 102, 72, 99, 72, 36, 117, 90, 90, 123, 108, 108, 81, 88, 126, 116, 135, 102, 48

Offset: 1

Juri-Stepan Gerasimov, Aug 14 2008

nonn

In the prime number decomposition of k=A002808(i), i=1,2,3,.., one instance of the largest prime, pmax=A052369(i), is replaced by pmax-2 and one instance of the smallest prime, pmin=A056608(i), is replaced by pmin+1. If the product of this modified list of factors, k*(pmax-2)*(pmin+1)/(pmin*pmax), is nonprime, it is added to the sequence.

			k(1)=(p(max)=2)*(p(min)=2), transformed (2-2)*(2+1)=0*3=0=a(1).
k(2)=(p(max)=3)*(p(min)=2), transformed (3-2)*(2+1)=1*3=3 (prime, skipped).
k(3)=(p(max)=2)*(p=2)*(p(min)=2), transformed (2-2)*2*(2+1)=0*2*3=0=a(2), etc.

Cf. A141218, A141219, A141220, A141284.

Edited and corrected by R. J. Mathar, Aug 18 2008

cp's OEIS Frontend

A141284 a(n) = (p_max - 1)*...*p*...*(p_min + 2), where p_max*...*p*...*p_min = k(n) = n-th composite.

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A161986 a(n) = k+r where k is composite(n) and r is (largest prime divisor of k) mod (smallest prime divisor of k).

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A082049 Least composite number greater than n-th composite number having greater smallest prime factor than that of n-th composite number.

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A118663 Index of the least prime dividing the n-th composite number: A000720(A020639(A002808(n))).

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A141554 Transformed nonprime products of prime factors of the composites, the largest prime decremented by 2 and the smallest prime incremented by 2.

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A161850 Subsequence of A161986 consisting of all terms that are prime.

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Magma

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A242016 Smallest prime factor of A135972(n), the n-th composite Mersenne number.

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A085434 Twice odd isolated primes.

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A141284 a(n) = (p_max - 1)...p...(p_min + 2), where p_max...p...p_min = k(n) = n-th composite.